3.506 \(\int \frac{x^6}{\left (a+b x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=91 \[ -\frac{5 a \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{2 b^{7/2}}+\frac{5 x \sqrt{a+b x^2}}{2 b^3}-\frac{5 x^3}{3 b^2 \sqrt{a+b x^2}}-\frac{x^5}{3 b \left (a+b x^2\right )^{3/2}} \]

[Out]

-x^5/(3*b*(a + b*x^2)^(3/2)) - (5*x^3)/(3*b^2*Sqrt[a + b*x^2]) + (5*x*Sqrt[a + b
*x^2])/(2*b^3) - (5*a*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(2*b^(7/2))

_______________________________________________________________________________________

Rubi [A]  time = 0.099484, antiderivative size = 91, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267 \[ -\frac{5 a \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{2 b^{7/2}}+\frac{5 x \sqrt{a+b x^2}}{2 b^3}-\frac{5 x^3}{3 b^2 \sqrt{a+b x^2}}-\frac{x^5}{3 b \left (a+b x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[x^6/(a + b*x^2)^(5/2),x]

[Out]

-x^5/(3*b*(a + b*x^2)^(3/2)) - (5*x^3)/(3*b^2*Sqrt[a + b*x^2]) + (5*x*Sqrt[a + b
*x^2])/(2*b^3) - (5*a*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(2*b^(7/2))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 13.6176, size = 83, normalized size = 0.91 \[ - \frac{5 a \operatorname{atanh}{\left (\frac{\sqrt{b} x}{\sqrt{a + b x^{2}}} \right )}}{2 b^{\frac{7}{2}}} - \frac{x^{5}}{3 b \left (a + b x^{2}\right )^{\frac{3}{2}}} - \frac{5 x^{3}}{3 b^{2} \sqrt{a + b x^{2}}} + \frac{5 x \sqrt{a + b x^{2}}}{2 b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**6/(b*x**2+a)**(5/2),x)

[Out]

-5*a*atanh(sqrt(b)*x/sqrt(a + b*x**2))/(2*b**(7/2)) - x**5/(3*b*(a + b*x**2)**(3
/2)) - 5*x**3/(3*b**2*sqrt(a + b*x**2)) + 5*x*sqrt(a + b*x**2)/(2*b**3)

_______________________________________________________________________________________

Mathematica [A]  time = 0.148745, size = 73, normalized size = 0.8 \[ \frac{15 a^2 x+20 a b x^3+3 b^2 x^5}{6 b^3 \left (a+b x^2\right )^{3/2}}-\frac{5 a \log \left (\sqrt{b} \sqrt{a+b x^2}+b x\right )}{2 b^{7/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^6/(a + b*x^2)^(5/2),x]

[Out]

(15*a^2*x + 20*a*b*x^3 + 3*b^2*x^5)/(6*b^3*(a + b*x^2)^(3/2)) - (5*a*Log[b*x + S
qrt[b]*Sqrt[a + b*x^2]])/(2*b^(7/2))

_______________________________________________________________________________________

Maple [A]  time = 0.013, size = 75, normalized size = 0.8 \[{\frac{{x}^{5}}{2\,b} \left ( b{x}^{2}+a \right ) ^{-{\frac{3}{2}}}}+{\frac{5\,a{x}^{3}}{6\,{b}^{2}} \left ( b{x}^{2}+a \right ) ^{-{\frac{3}{2}}}}+{\frac{5\,ax}{2\,{b}^{3}}{\frac{1}{\sqrt{b{x}^{2}+a}}}}-{\frac{5\,a}{2}\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ){b}^{-{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^6/(b*x^2+a)^(5/2),x)

[Out]

1/2*x^5/b/(b*x^2+a)^(3/2)+5/6*a/b^2*x^3/(b*x^2+a)^(3/2)+5/2*a/b^3*x/(b*x^2+a)^(1
/2)-5/2*a/b^(7/2)*ln(x*b^(1/2)+(b*x^2+a)^(1/2))

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^6/(b*x^2 + a)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.255383, size = 1, normalized size = 0.01 \[ \left [\frac{2 \,{\left (3 \, b^{2} x^{5} + 20 \, a b x^{3} + 15 \, a^{2} x\right )} \sqrt{b x^{2} + a} \sqrt{b} + 15 \,{\left (a b^{2} x^{4} + 2 \, a^{2} b x^{2} + a^{3}\right )} \log \left (2 \, \sqrt{b x^{2} + a} b x -{\left (2 \, b x^{2} + a\right )} \sqrt{b}\right )}{12 \,{\left (b^{5} x^{4} + 2 \, a b^{4} x^{2} + a^{2} b^{3}\right )} \sqrt{b}}, \frac{{\left (3 \, b^{2} x^{5} + 20 \, a b x^{3} + 15 \, a^{2} x\right )} \sqrt{b x^{2} + a} \sqrt{-b} - 15 \,{\left (a b^{2} x^{4} + 2 \, a^{2} b x^{2} + a^{3}\right )} \arctan \left (\frac{\sqrt{-b} x}{\sqrt{b x^{2} + a}}\right )}{6 \,{\left (b^{5} x^{4} + 2 \, a b^{4} x^{2} + a^{2} b^{3}\right )} \sqrt{-b}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^6/(b*x^2 + a)^(5/2),x, algorithm="fricas")

[Out]

[1/12*(2*(3*b^2*x^5 + 20*a*b*x^3 + 15*a^2*x)*sqrt(b*x^2 + a)*sqrt(b) + 15*(a*b^2
*x^4 + 2*a^2*b*x^2 + a^3)*log(2*sqrt(b*x^2 + a)*b*x - (2*b*x^2 + a)*sqrt(b)))/((
b^5*x^4 + 2*a*b^4*x^2 + a^2*b^3)*sqrt(b)), 1/6*((3*b^2*x^5 + 20*a*b*x^3 + 15*a^2
*x)*sqrt(b*x^2 + a)*sqrt(-b) - 15*(a*b^2*x^4 + 2*a^2*b*x^2 + a^3)*arctan(sqrt(-b
)*x/sqrt(b*x^2 + a)))/((b^5*x^4 + 2*a*b^4*x^2 + a^2*b^3)*sqrt(-b))]

_______________________________________________________________________________________

Sympy [A]  time = 16.6398, size = 367, normalized size = 4.03 \[ - \frac{15 a^{\frac{81}{2}} b^{22} \sqrt{1 + \frac{b x^{2}}{a}} \operatorname{asinh}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{6 a^{\frac{79}{2}} b^{\frac{51}{2}} \sqrt{1 + \frac{b x^{2}}{a}} + 6 a^{\frac{77}{2}} b^{\frac{53}{2}} x^{2} \sqrt{1 + \frac{b x^{2}}{a}}} - \frac{15 a^{\frac{79}{2}} b^{23} x^{2} \sqrt{1 + \frac{b x^{2}}{a}} \operatorname{asinh}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{6 a^{\frac{79}{2}} b^{\frac{51}{2}} \sqrt{1 + \frac{b x^{2}}{a}} + 6 a^{\frac{77}{2}} b^{\frac{53}{2}} x^{2} \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{15 a^{40} b^{\frac{45}{2}} x}{6 a^{\frac{79}{2}} b^{\frac{51}{2}} \sqrt{1 + \frac{b x^{2}}{a}} + 6 a^{\frac{77}{2}} b^{\frac{53}{2}} x^{2} \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{20 a^{39} b^{\frac{47}{2}} x^{3}}{6 a^{\frac{79}{2}} b^{\frac{51}{2}} \sqrt{1 + \frac{b x^{2}}{a}} + 6 a^{\frac{77}{2}} b^{\frac{53}{2}} x^{2} \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{3 a^{38} b^{\frac{49}{2}} x^{5}}{6 a^{\frac{79}{2}} b^{\frac{51}{2}} \sqrt{1 + \frac{b x^{2}}{a}} + 6 a^{\frac{77}{2}} b^{\frac{53}{2}} x^{2} \sqrt{1 + \frac{b x^{2}}{a}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**6/(b*x**2+a)**(5/2),x)

[Out]

-15*a**(81/2)*b**22*sqrt(1 + b*x**2/a)*asinh(sqrt(b)*x/sqrt(a))/(6*a**(79/2)*b**
(51/2)*sqrt(1 + b*x**2/a) + 6*a**(77/2)*b**(53/2)*x**2*sqrt(1 + b*x**2/a)) - 15*
a**(79/2)*b**23*x**2*sqrt(1 + b*x**2/a)*asinh(sqrt(b)*x/sqrt(a))/(6*a**(79/2)*b*
*(51/2)*sqrt(1 + b*x**2/a) + 6*a**(77/2)*b**(53/2)*x**2*sqrt(1 + b*x**2/a)) + 15
*a**40*b**(45/2)*x/(6*a**(79/2)*b**(51/2)*sqrt(1 + b*x**2/a) + 6*a**(77/2)*b**(5
3/2)*x**2*sqrt(1 + b*x**2/a)) + 20*a**39*b**(47/2)*x**3/(6*a**(79/2)*b**(51/2)*s
qrt(1 + b*x**2/a) + 6*a**(77/2)*b**(53/2)*x**2*sqrt(1 + b*x**2/a)) + 3*a**38*b**
(49/2)*x**5/(6*a**(79/2)*b**(51/2)*sqrt(1 + b*x**2/a) + 6*a**(77/2)*b**(53/2)*x*
*2*sqrt(1 + b*x**2/a))

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.235957, size = 88, normalized size = 0.97 \[ \frac{{\left (x^{2}{\left (\frac{3 \, x^{2}}{b} + \frac{20 \, a}{b^{2}}\right )} + \frac{15 \, a^{2}}{b^{3}}\right )} x}{6 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}}} + \frac{5 \, a{\rm ln}\left ({\left | -\sqrt{b} x + \sqrt{b x^{2} + a} \right |}\right )}{2 \, b^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^6/(b*x^2 + a)^(5/2),x, algorithm="giac")

[Out]

1/6*(x^2*(3*x^2/b + 20*a/b^2) + 15*a^2/b^3)*x/(b*x^2 + a)^(3/2) + 5/2*a*ln(abs(-
sqrt(b)*x + sqrt(b*x^2 + a)))/b^(7/2)